Crystal-field approximation

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 729-730

Section 8.7.4.5.1.2. Crystal-field approximation

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,^bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and ^cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France

8.7.4.5.1.2. Crystal-field approximation

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Crystal-field effects are generally of major importance in spin magnetism and are responsible for the spin state of the ions, and thus for the ground-state configuration of the system. Thus, they have to be incorporated in the model.

Taking the case of a transition-metal compound, and neglecting small contributions that may arise from spin polarization in the closed shells (see Subsections 8.7.4.9 and 8.7.4.10), the normalized spin density can be written by analogy with (8.7.3.76) as $[s({\bf r}) = \textstyle\sum\limits^5_{i=1}\, \textstyle\sum\limits^5_{j\ge i}\,D_{ij}\, d_i ({\bf r}) d_j ({\bf r}), \eqno (8.7.4.57)]$ where $[D_{ij}]$ is the normalized spin population matrix. If $[\rho_{d\uparrow}]$ and $[\rho_{d\downarrow}]$ are the densities of a given spin, $[s({\bf r}) = {\rho_{d\uparrow}-\rho_{d\downarrow} \over n_\uparrow - n_\downarrow}, \eqno (8.7.4.58)]$ the d-type charge density is $[\rho_d({\bf r})=\rho_{d\uparrow}+\rho_{d\downarrow} \eqno (8.7.4.59)]$ and is expanded in a similar way to s(r) [see (8.7.3.76)], $[\rho_d({\bf r})=\textstyle\sum\limits_i\, \textstyle\sum\limits_{j\ge i}\,P_{ij} d_i d_j\semi \eqno (8.7.4.60)]$ writing $[\rho_{d \sigma} = \textstyle\sum\, P^\sigma_{ij}\, d_i d_j \eqno (8.7.4.61)]$ with σ = $[ \uparrow]$ and $[\downarrow]$ , one obtains $[\matrix{ P_{ij} = P^\uparrow_{ij} +P^\downarrow_{ij} \cr \vphantom{}\cr D_{ij} = (P^\uparrow_{ij} - P^\downarrow_{ij})/(n_\uparrow - n_\downarrow).} \eqno (8.7.4.62)]$ Similarly to $[\rho_d({\bf r})]$ , [s(r)] can be expanded as $[s({\bf r})= U^2 (r)\, \textstyle\sum\limits^4_{l=0}\; \textstyle\sum\limits^l_{m=0}\; \textstyle\sum\limits_p\, D_{lmp}\, y_{lmp}(\theta, \varphi), \eqno (8.7.4.63)]$ where U(r) describes the radial dependence. Spin polarization leads to a further modification of this expression. Since the numbers of electrons of a given spin are different, the exchange interaction is different for the two spin states, and a spin-dependent effective screening occurs. This leads to $[\rho_{d\sigma}({\bf r}) = \kappa^3_\sigma U^2(\kappa_\sigma r)\textstyle \sum\limits_{lmp}\, P^\sigma_{lmp}\, y_{lmp}, \eqno (8.7.4.64)]$ with σ = $[\ \uparrow]$ or $[\downarrow]$ , where two κ parameters are needed.

The complementarity between charge and spin density in the crystal field approximation is obvious. At this particular level of approximation, expansions are exact and it is possible to estimate d-orbital populations for each spin state.

References

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 729-730